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2-d: images

Let us assume we have a continuous distribution, on a plane, of values of luminance or, more simply stated, animage. In order to process it using a computer we have to reduce it to a sequence of numbers by means ofsampling. There are several ways to sample an image, or read its values of luminance at discrete points. Thesimplest way is to use a regular grid, with spatial steps X e Y . Similarly to what we did for sounds, we define the spatial sampling rates F X 1 X and F Y 1 Y . As in the one-dimensional case, also for two-dimensional signals, or images, sampling can bedescribed by three facts and a theorem.

  • The Fourier Transform of a discrete-space signal is a function (called spectrum ) of two continuous variables ω X and ω Y , and it is periodic in two dimensions with periods 2 π . Given a couple of values ω X and ω Y , the Fourier transform gives back a complex number that can be interpreted as magnitude andphase (translation in space) of the sinusoidal component at such spatial frequencies.
  • Sampling the continuous-space signal s x y with the regular grid of steps X , Y , gives a discrete-space signal s m n s m X n Y , which is a function of the discrete variables m and n .
  • Sampling a continuous-space signal with spatial frequencies F X and F Y gives a discrete-space signal whose spectrum is the periodic replication along the grid of steps F X and F Y of the original signal spectrum. The Fourier variables ω X and ω Y correspond to the frequencies (in cycles per meter) represented by the variables f X ω X 2 π X and f Y ω Y 2 π Y .

The [link] shows an example of spectrum of a two-dimensional sampled signal. There, thecontinuous-space signal had all and only the frequency components included in the central hexagon. The hexagonalshape of the spectral support (region of non-null spectral energy) is merely illustrative. The replicas of the originalspectrum are often called spectral images .

Spectrum of a sampled image

Given the above facts , we can have an intuitive understanding of the Sampling Theorem.

Sampling theorem (in 2d)

A continuous-space signal s x y , whose spectral content is limited to spatial frequencies belonging to the rectangle having semi-edges F bX and F bY (i.e., bandlimited) can be recovered from its sampled version s m n if the spatial sampling rates are larger than twice the respective bandwidths (i.e., if F X 2 F bX and F Y 2 F bY )

In practice, the spatial sampling step can not be larger than the semi-period of the finest spatial frequency (or thefinest detail) that is represented in the image. The reconstruction can only be done through a filter thateliminates all the spectral images but the one coming directly from the original continuous-space signal. In otherwords, the filter will cut all images whose frequency components are higher than the Nyquist frequency defined as F X 2 and F Y 2 along the two axes. The condition required by the sampling theorem is equivalent to requiring that there are no overlaps between spectral images. If there were suchoverlaps, it wouldn't be possible to eliminate the copies of the original signal spectrum by means of filtering. In case of overlapping, a filtercutting all frequency components higher than the Nyquist frequency would give back a signal that is affected byaliasing.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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I'm interested in nanotube
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Source:  OpenStax, Media processing in processing. OpenStax CNX. Nov 10, 2010 Download for free at http://cnx.org/content/col10268/1.14
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